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Chapter 5: Q 24. (page 452)

Solve the integral.
$鈭� \sin^{3} x d x$

Short Answer

Expert verified

Answer is $\begin{array}{rcl} \frac{\cos^{3} x}{3} - \cos x + C \end{array}$

Step by step solution

01

Step 1. Given information

An integral is $鈭� \sin^{3} x d x$

02

Step 2. Explanation

Consider the integral $鈭� \sin^{3} x d x$

$\begin{array}{rcl} 鈭� \sin^{3} x d x & = & 鈭� \sin x \sin^{2} x d x \\ = & 鈭� \sin x (1 - \cos^{2} x) d x \end{array}$

Let $\cos x = u$

$- \sin x d x = d u$

So,

$\begin{array}{rcl} 鈭� \sin^{3} x d x & = & 鈭� u^{2} - 1 d u \\ = & \frac{u^{3}}{3} - u + C \\ = & \frac{\cos^{3} x}{3} - \cos x + C \end{array}$

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Most popular questions from this chapter

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $鈭� \frac{1}{x^{2} 鈭� 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $鈭� \frac{1}{\sqrt{x^{2} 鈭� 4}} d x$ .

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $鈭� \frac{1}{x^{2} + 4} d x .$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $鈭� {(x^{2} + 4)}^{鈭� 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} 鈭� a^{2}}$ .

(f) True or False: Trigonometric substitution doesn鈥檛 solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� x^{3} \sqrt{x^{2} + 1} d x$

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Solve the integral

$鈭� \frac{1}{x^{2} \sqrt{x^{2} - 9}} d x$

Solve the integral: $鈭� x \ln \sqrt{x} d x$

See all solutions

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